feat(LinearAlgebra/Transvection): the determinant of a transvection is equal to 1.

[AntoineChambert-Loir]

Prove that the determinant of a transvection is equal to 1

The proof goes by showing that the determinant of LinearMap.transvection f v is 1 + f v (even if f v is nonzero).

I first treat the case of a field, distinguishing whether f v = 0 (so that we get a transvection) or not (so that we have a dilation).

Then, by base change to the field of fractions, I can handle the case of a domain. Finally, the general case is treated by base change from the “universal case” where the base ring is the ring of polynomials in $$2n$$ indeterminates corresponding to the coefficients of $$f$$ and $$v$$.

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• [ ] depends on: #31138
• [ ]

[github-actions[bot]]

PR summary ff7ce573c0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ _root_.LinearEquiv.det_eq_one + _root_.Module.Free.of_det_ne_one + det + det_ofDomain + det_ofField + exists_basis_of_pairing_eq_zero + exists_basis_of_pairing_ne_zero

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[github-actions[bot]]

✅ PR Title Formatted Correctly

The title of this PR has been updated to match our commit style conventions. Thank you!

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• ~~leanprover-community/mathlib4#31138~~ By Dependent Issues (🤖). Happy coding!

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.